Nested arabic enumeration without last dot

Question

I like to change the labeling of an enumeration to the following structure:

1 First Item 
  1.1 First Sub-Item 
    1.1.1 First Subsub-Item  
    1.1.2 Second Subsub-Item  
  1.2 Second Sub-Item   
2 Second Item   
3 Third Item 

Similiar questions have already been asked and the following code using label*=\arabic*. (and which is also part of the official enumitem documenation) is always the accepted answer:

\documentclass{article}
\usepackage{enumitem}
\usepackage{enumitem}
\newlist{legal}{enumerate}{10}
\setlist[legal]{label*=\arabic*.}

\begin{document}
\begin{legal}
  \item First Item 
  \begin{legal}
    \item First Sub-Item 
      \begin{legal}
        \item First Subsub-Item 
        \item Second Subsub-Item 
      \end{legal}
    \item Second Sub-Item 
  \end{legal}
  \item Second Item 
  \item Third Item 
\end{legal}
\end{document}

Unfortunately this solution puts an dot . after the last level of enumeration and therefore gives the following result:

1. First Item   
  1.1. First Sub-Item  
    1.1.1. First Subsub-Item  
    1.1.2. Second Subsub-Item   
  1.2. Second Sub-Item   
2. Second Item   
3. Third Item 

Is it possible to define the label in such a way that it is not [1.1., 1.1.1., etc.] but instead [1.1, 1.1.1, etc.]?

Answer 1

Implementing @moewe's comments, here's a solution that requires only two separate \setlist instructions. The first \setlist instructions pertains to all list levels; the second affects just the numeric labels of level-1 items.

In addition, if you would rather not have any extra vertical whitespace between any of the items, simply provide the instruction \setlist[legal]{nosep} after the \newlist directive.

\documentclass{article}
\usepackage{enumitem}
\newlist{legal}{enumerate}{10}
\setlist[legal]{label*=.\arabic*}
\setlist[legal,1]{label=\arabic*}

\begin{document}
\begin{legal}
  \item First Item 
  \begin{legal}
    \item First Sub-Item 
      \begin{legal}
        \item First Subsub-Item 
        \item Second Subsub-Item 
          \begin{legal}
            \item First Subsubsub-Item
            \item Second Subsubsub-Item
          \end{legal}
        \item Third Subsub-Item
      \end{legal}
    \item Second Sub-Item 
  \end{legal}
  \item Second Item 
\end{legal}
\end{document}